9 3

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9 2.gif

9_2

9 4.gif

9_4

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9 3 Quick Notes


9 3 Further Notes and Views

Knot presentations

Planar diagram presentation X8291 X12,4,13,3 X18,10,1,9 X10,18,11,17 X14,6,15,5 X16,8,17,7 X2,12,3,11 X4,14,5,13 X6,16,7,15
Gauss code 1, -7, 2, -8, 5, -9, 6, -1, 3, -4, 7, -2, 8, -5, 9, -6, 4, -3
Dowker-Thistlethwaite code 8 12 14 16 18 2 4 6 10
Conway Notation [63]

Three dimensional invariants

Symmetry type Reversible
Unknotting number 3
3-genus 3
Bridge index 2
Super bridge index [math]\displaystyle{ \{4,6\} }[/math]
Nakanishi index 1
Maximal Thurston-Bennequin number [5][-16]
Hyperbolic Volume 4.99486
A-Polynomial See Data:9 3/A-polynomial

[edit Notes for 9 3's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus [math]\displaystyle{ 3 }[/math]
Topological 4 genus [math]\displaystyle{ 3 }[/math]
Concordance genus [math]\displaystyle{ 3 }[/math]
Rasmussen s-Invariant -6

[edit Notes for 9 3's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ 2 t^3-3 t^2+3 t-3+3 t^{-1} -3 t^{-2} +2 t^{-3} }[/math]
Conway polynomial [math]\displaystyle{ 2 z^6+9 z^4+9 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 19, 6 }
Jones polynomial [math]\displaystyle{ -q^{12}+q^{11}-2 q^{10}+3 q^9-3 q^8+3 q^7-2 q^6+2 q^5-q^4+q^3 }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ z^6 a^{-6} +z^6 a^{-8} +5 z^4 a^{-6} +5 z^4 a^{-8} -z^4 a^{-10} +6 z^2 a^{-6} +7 z^2 a^{-8} -4 z^2 a^{-10} + a^{-6} +3 a^{-8} -3 a^{-10} }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^8 a^{-8} +z^8 a^{-10} +z^7 a^{-7} +2 z^7 a^{-9} +z^7 a^{-11} +z^6 a^{-6} -5 z^6 a^{-8} -5 z^6 a^{-10} +z^6 a^{-12} -4 z^5 a^{-7} -8 z^5 a^{-9} -3 z^5 a^{-11} +z^5 a^{-13} -5 z^4 a^{-6} +9 z^4 a^{-8} +11 z^4 a^{-10} -2 z^4 a^{-12} +z^4 a^{-14} +3 z^3 a^{-7} +9 z^3 a^{-9} +4 z^3 a^{-11} -z^3 a^{-13} +z^3 a^{-15} +6 z^2 a^{-6} -9 z^2 a^{-8} -11 z^2 a^{-10} +3 z^2 a^{-12} -z^2 a^{-14} -4 z a^{-9} -z a^{-11} +z a^{-13} -2 z a^{-15} - a^{-6} +3 a^{-8} +3 a^{-10} }[/math]
The A2 invariant [math]\displaystyle{ q^{-10} + q^{-14} + q^{-18} + q^{-20} + q^{-22} +2 q^{-24} - q^{-30} - q^{-32} - q^{-34} - q^{-36} }[/math]
The G2 invariant [math]\displaystyle{ q^{-50} + q^{-54} - q^{-56} + q^{-58} +3 q^{-64} -2 q^{-66} +3 q^{-68} - q^{-70} + q^{-72} +2 q^{-74} -3 q^{-76} +3 q^{-78} - q^{-80} + q^{-82} + q^{-84} - q^{-86} +2 q^{-88} + q^{-90} +2 q^{-94} - q^{-96} + q^{-98} +2 q^{-100} -2 q^{-102} +3 q^{-104} - q^{-106} +2 q^{-108} - q^{-112} +2 q^{-114} -2 q^{-116} +3 q^{-118} -2 q^{-120} + q^{-124} -2 q^{-126} + q^{-128} - q^{-130} - q^{-132} + q^{-134} - q^{-136} - q^{-138} - q^{-142} - q^{-146} -2 q^{-148} - q^{-152} - q^{-156} - q^{-160} - q^{-162} - q^{-164} - q^{-166} +2 q^{-168} -2 q^{-170} + q^{-172} + q^{-178} - q^{-180} + q^{-182} - q^{-184} + q^{-186} - q^{-190} + q^{-192} + q^{-196} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 9_3.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ q^{-5} + q^{-9} + q^{-13} + q^{-19} - q^{-21} - q^{-25} }[/math]
2 [math]\displaystyle{ q^{-10} +2 q^{-16} + q^{-18} - q^{-20} + q^{-22} + q^{-24} - q^{-26} + q^{-30} + q^{-36} - q^{-40} - q^{-48} + q^{-50} - q^{-52} -2 q^{-54} - q^{-58} + q^{-62} + q^{-68} }[/math]
3 [math]\displaystyle{ q^{-15} + q^{-21} +2 q^{-23} + q^{-25} - q^{-27} - q^{-29} +2 q^{-31} +2 q^{-33} -2 q^{-37} +2 q^{-41} + q^{-43} - q^{-45} - q^{-47} + q^{-51} + q^{-53} - q^{-57} +2 q^{-61} -2 q^{-65} - q^{-67} + q^{-69} - q^{-71} -2 q^{-73} + q^{-77} - q^{-81} - q^{-83} - q^{-85} + q^{-87} + q^{-89} -2 q^{-91} - q^{-93} - q^{-95} +2 q^{-97} - q^{-101} - q^{-103} + q^{-105} +2 q^{-107} + q^{-109} - q^{-111} +2 q^{-115} + q^{-117} - q^{-121} - q^{-129} }[/math]
4 [math]\displaystyle{ q^{-20} + q^{-26} + q^{-28} +2 q^{-30} - q^{-34} +4 q^{-40} +2 q^{-42} - q^{-44} -2 q^{-46} -3 q^{-48} +2 q^{-50} +3 q^{-52} +2 q^{-54} -4 q^{-58} - q^{-60} + q^{-62} +2 q^{-64} +2 q^{-66} - q^{-68} - q^{-72} - q^{-74} + q^{-76} + q^{-78} +3 q^{-80} -4 q^{-84} -3 q^{-86} +4 q^{-90} +2 q^{-92} -4 q^{-94} -4 q^{-96} - q^{-98} +4 q^{-100} +3 q^{-102} -3 q^{-104} -4 q^{-106} - q^{-108} +2 q^{-110} +2 q^{-112} -2 q^{-114} -2 q^{-116} + q^{-120} + q^{-122} -2 q^{-124} -2 q^{-126} + q^{-130} + q^{-132} + q^{-134} - q^{-136} -3 q^{-138} -2 q^{-140} +2 q^{-142} +6 q^{-144} +3 q^{-146} - q^{-148} -7 q^{-150} -3 q^{-152} +6 q^{-154} +6 q^{-156} +2 q^{-158} -7 q^{-160} -4 q^{-162} +3 q^{-164} +5 q^{-166} +4 q^{-168} -3 q^{-170} -4 q^{-172} + q^{-174} +2 q^{-176} +3 q^{-178} -2 q^{-180} -3 q^{-182} - q^{-184} +2 q^{-188} - q^{-190} - q^{-192} - q^{-194} - q^{-196} + q^{-198} + q^{-208} }[/math]
5 [math]\displaystyle{ q^{-25} + q^{-31} + q^{-33} + q^{-35} + q^{-37} - q^{-41} +2 q^{-45} +2 q^{-47} +3 q^{-49} + q^{-51} -2 q^{-53} -4 q^{-55} -2 q^{-57} + q^{-59} +4 q^{-61} +5 q^{-63} +2 q^{-65} -2 q^{-67} -5 q^{-69} -4 q^{-71} +3 q^{-75} +5 q^{-77} +3 q^{-79} - q^{-81} -4 q^{-83} -3 q^{-85} + q^{-89} +2 q^{-91} +2 q^{-93} + q^{-95} + q^{-97} + q^{-99} -2 q^{-101} -4 q^{-103} -4 q^{-105} +5 q^{-109} +6 q^{-111} +2 q^{-113} -5 q^{-115} -10 q^{-117} -6 q^{-119} +3 q^{-121} +9 q^{-123} +7 q^{-125} - q^{-127} -9 q^{-129} -10 q^{-131} -3 q^{-133} +7 q^{-135} +10 q^{-137} +4 q^{-139} -4 q^{-141} -9 q^{-143} -6 q^{-145} +2 q^{-147} +8 q^{-149} +5 q^{-151} -3 q^{-153} -7 q^{-155} -7 q^{-157} +6 q^{-161} +5 q^{-163} -4 q^{-167} -3 q^{-169} + q^{-171} +3 q^{-173} +3 q^{-175} -2 q^{-177} -3 q^{-179} +2 q^{-183} +2 q^{-185} + q^{-187} - q^{-189} -2 q^{-191} -2 q^{-193} +2 q^{-197} +4 q^{-199} +6 q^{-201} +2 q^{-203} -5 q^{-205} -8 q^{-207} -4 q^{-209} +3 q^{-211} +11 q^{-213} +13 q^{-215} - q^{-217} -12 q^{-219} -13 q^{-221} -3 q^{-223} +9 q^{-225} +15 q^{-227} +5 q^{-229} -8 q^{-231} -11 q^{-233} -6 q^{-235} +5 q^{-237} +8 q^{-239} +3 q^{-241} -4 q^{-243} -6 q^{-245} -3 q^{-247} +3 q^{-249} +4 q^{-251} + q^{-253} -4 q^{-255} -3 q^{-257} - q^{-259} +2 q^{-261} +2 q^{-263} + q^{-265} -2 q^{-267} -3 q^{-269} - q^{-271} + q^{-273} + q^{-275} +2 q^{-277} + q^{-279} - q^{-281} + q^{-289} + q^{-291} - q^{-305} }[/math]
6 [math]\displaystyle{ q^{-30} + q^{-36} + q^{-38} + q^{-40} + q^{-44} - q^{-48} + q^{-50} +2 q^{-52} +3 q^{-54} + q^{-56} +2 q^{-58} - q^{-60} -4 q^{-62} -3 q^{-64} - q^{-66} +3 q^{-68} +3 q^{-70} +7 q^{-72} +4 q^{-74} -2 q^{-76} -5 q^{-78} -7 q^{-80} -4 q^{-82} -2 q^{-84} +6 q^{-86} +8 q^{-88} +6 q^{-90} +2 q^{-92} -3 q^{-94} -6 q^{-96} -9 q^{-98} -2 q^{-100} +2 q^{-102} +6 q^{-104} +6 q^{-106} +3 q^{-108} + q^{-110} -4 q^{-112} -2 q^{-114} -2 q^{-116} - q^{-120} -2 q^{-122} +5 q^{-128} +6 q^{-130} +5 q^{-132} -3 q^{-134} -11 q^{-136} -11 q^{-138} -9 q^{-140} +2 q^{-142} +12 q^{-144} +17 q^{-146} +9 q^{-148} -5 q^{-150} -15 q^{-152} -20 q^{-154} -11 q^{-156} +4 q^{-158} +18 q^{-160} +19 q^{-162} +9 q^{-164} -3 q^{-166} -18 q^{-168} -20 q^{-170} -11 q^{-172} +5 q^{-174} +16 q^{-176} +17 q^{-178} +12 q^{-180} -5 q^{-182} -18 q^{-184} -19 q^{-186} -9 q^{-188} +5 q^{-190} +16 q^{-192} +20 q^{-194} +7 q^{-196} -10 q^{-198} -19 q^{-200} -15 q^{-202} -3 q^{-204} +10 q^{-206} +20 q^{-208} +11 q^{-210} -5 q^{-212} -14 q^{-214} -12 q^{-216} -3 q^{-218} +7 q^{-220} +16 q^{-222} +10 q^{-224} -4 q^{-226} -10 q^{-228} -8 q^{-230} -2 q^{-232} +4 q^{-234} +8 q^{-236} +4 q^{-238} -4 q^{-240} -4 q^{-242} +2 q^{-246} +3 q^{-248} +3 q^{-250} -3 q^{-254} -3 q^{-256} +3 q^{-260} +4 q^{-262} +4 q^{-264} +2 q^{-266} + q^{-268} -4 q^{-270} -8 q^{-272} -7 q^{-274} -2 q^{-276} +4 q^{-278} +12 q^{-280} +15 q^{-282} +6 q^{-284} -9 q^{-286} -18 q^{-288} -17 q^{-290} -10 q^{-292} +12 q^{-294} +25 q^{-296} +22 q^{-298} +5 q^{-300} -15 q^{-302} -24 q^{-304} -25 q^{-306} -3 q^{-308} +15 q^{-310} +22 q^{-312} +13 q^{-314} - q^{-316} -12 q^{-318} -18 q^{-320} -7 q^{-322} +7 q^{-326} +7 q^{-328} +4 q^{-330} +3 q^{-332} -3 q^{-334} -3 q^{-338} -4 q^{-340} -4 q^{-342} +6 q^{-346} +3 q^{-348} +6 q^{-350} -4 q^{-354} -6 q^{-356} -3 q^{-358} +2 q^{-360} +2 q^{-362} +6 q^{-364} +3 q^{-366} -3 q^{-370} -3 q^{-372} - q^{-374} - q^{-376} +3 q^{-378} +2 q^{-380} +2 q^{-382} - q^{-388} -2 q^{-390} - q^{-394} + q^{-396} - q^{-404} - q^{-408} + q^{-420} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{-10} + q^{-14} + q^{-18} + q^{-20} + q^{-22} +2 q^{-24} - q^{-30} - q^{-32} - q^{-34} - q^{-36} }[/math]
1,1 [math]\displaystyle{ q^{-20} +2 q^{-24} -2 q^{-26} +6 q^{-28} -2 q^{-30} +10 q^{-32} -4 q^{-34} +9 q^{-36} -2 q^{-38} +4 q^{-40} -4 q^{-44} +6 q^{-46} -8 q^{-48} +8 q^{-50} -11 q^{-52} +8 q^{-54} -10 q^{-56} +4 q^{-58} -8 q^{-60} -2 q^{-64} -2 q^{-66} +2 q^{-68} +2 q^{-72} +2 q^{-74} + q^{-76} -2 q^{-78} -2 q^{-82} +3 q^{-84} -4 q^{-86} +2 q^{-88} -2 q^{-90} +2 q^{-92} -2 q^{-94} +2 q^{-96} + q^{-100} }[/math]
2,0 [math]\displaystyle{ q^{-20} + q^{-26} +2 q^{-28} + q^{-30} + q^{-32} +2 q^{-34} +2 q^{-36} + q^{-42} + q^{-44} + q^{-46} +2 q^{-48} + q^{-50} - q^{-58} -3 q^{-66} -3 q^{-68} -3 q^{-70} -3 q^{-72} -3 q^{-74} -2 q^{-76} + q^{-80} +2 q^{-82} + q^{-84} +2 q^{-86} + q^{-88} + q^{-90} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{-20} + q^{-24} + q^{-26} + q^{-30} +2 q^{-32} +2 q^{-34} +4 q^{-36} +3 q^{-38} +2 q^{-40} +2 q^{-42} + q^{-44} -2 q^{-46} - q^{-48} -2 q^{-50} -3 q^{-52} -3 q^{-54} -2 q^{-56} - q^{-58} - q^{-60} + q^{-64} - q^{-66} - q^{-68} + q^{-70} - q^{-72} - q^{-74} + q^{-76} + q^{-78} + q^{-82} }[/math]
1,0,0 [math]\displaystyle{ q^{-15} + q^{-19} + q^{-23} +2 q^{-27} +2 q^{-29} +2 q^{-31} +2 q^{-33} -2 q^{-39} - q^{-41} -2 q^{-43} - q^{-45} - q^{-47} }[/math]
1,0,1 [math]\displaystyle{ q^{-30} +2 q^{-34} + q^{-38} +3 q^{-40} +2 q^{-42} +9 q^{-44} +5 q^{-46} +8 q^{-48} +7 q^{-50} +5 q^{-52} +7 q^{-54} +2 q^{-56} +5 q^{-58} - q^{-60} +4 q^{-62} -4 q^{-64} -2 q^{-66} -7 q^{-68} -12 q^{-70} -7 q^{-72} -14 q^{-74} -4 q^{-76} -9 q^{-78} + q^{-80} -3 q^{-82} +2 q^{-84} +3 q^{-86} -2 q^{-88} +6 q^{-90} -2 q^{-92} +6 q^{-94} + q^{-96} +3 q^{-98} +2 q^{-100} - q^{-102} -2 q^{-104} - q^{-106} + q^{-108} -2 q^{-110} +3 q^{-112} -2 q^{-114} - q^{-116} + q^{-118} -2 q^{-120} + q^{-122} - q^{-124} +2 q^{-128} + q^{-132} }[/math]

A4 Invariants.

Weight Invariant
0,1,0,0 [math]\displaystyle{ q^{-30} + q^{-34} + q^{-36} + q^{-38} +2 q^{-42} +2 q^{-44} +3 q^{-46} +4 q^{-48} +4 q^{-50} +5 q^{-52} +5 q^{-54} +4 q^{-56} +3 q^{-58} +3 q^{-60} +2 q^{-62} - q^{-64} -3 q^{-66} -4 q^{-68} -5 q^{-70} -7 q^{-72} -5 q^{-74} -4 q^{-76} -4 q^{-78} -2 q^{-80} - q^{-82} -2 q^{-84} -2 q^{-86} + q^{-94} +3 q^{-96} +2 q^{-98} + q^{-100} + q^{-102} + q^{-104} }[/math]
1,0,0,0 [math]\displaystyle{ q^{-20} + q^{-24} + q^{-28} + q^{-32} +2 q^{-34} +2 q^{-36} +3 q^{-38} +2 q^{-40} +2 q^{-42} -2 q^{-48} -2 q^{-50} -2 q^{-52} -2 q^{-54} - q^{-56} - q^{-58} }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ q^{-20} + q^{-24} - q^{-26} +2 q^{-28} - q^{-30} +2 q^{-32} +2 q^{-36} + q^{-38} +2 q^{-42} - q^{-44} +2 q^{-46} -3 q^{-48} +2 q^{-50} -3 q^{-52} + q^{-54} -2 q^{-56} + q^{-58} - q^{-60} + q^{-64} - q^{-66} + q^{-68} - q^{-70} + q^{-72} - q^{-74} + q^{-76} - q^{-78} - q^{-82} }[/math]
1,0 [math]\displaystyle{ q^{-30} + q^{-38} + q^{-40} - q^{-44} + q^{-46} +2 q^{-48} + q^{-50} - q^{-52} + q^{-54} +2 q^{-56} +3 q^{-58} + q^{-60} +2 q^{-66} + q^{-68} - q^{-72} - q^{-78} - q^{-80} - q^{-82} - q^{-86} -2 q^{-88} -2 q^{-90} - q^{-96} - q^{-98} + q^{-102} - q^{-106} - q^{-108} + q^{-112} - q^{-116} - q^{-118} + q^{-122} + q^{-124} + q^{-132} }[/math]

D4 Invariants.

Weight Invariant
1,0,0,0 [math]\displaystyle{ q^{-30} + q^{-34} +2 q^{-38} - q^{-40} +2 q^{-42} +3 q^{-46} +2 q^{-48} +3 q^{-50} +3 q^{-52} +4 q^{-54} +4 q^{-56} +2 q^{-58} +3 q^{-60} - q^{-62} + q^{-64} -4 q^{-66} - q^{-68} -5 q^{-70} -2 q^{-72} -5 q^{-74} - q^{-76} -3 q^{-78} - q^{-82} + q^{-90} - q^{-92} - q^{-96} + q^{-98} - q^{-100} - q^{-104} + q^{-106} + q^{-110} + q^{-114} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{-50} + q^{-54} - q^{-56} + q^{-58} +3 q^{-64} -2 q^{-66} +3 q^{-68} - q^{-70} + q^{-72} +2 q^{-74} -3 q^{-76} +3 q^{-78} - q^{-80} + q^{-82} + q^{-84} - q^{-86} +2 q^{-88} + q^{-90} +2 q^{-94} - q^{-96} + q^{-98} +2 q^{-100} -2 q^{-102} +3 q^{-104} - q^{-106} +2 q^{-108} - q^{-112} +2 q^{-114} -2 q^{-116} +3 q^{-118} -2 q^{-120} + q^{-124} -2 q^{-126} + q^{-128} - q^{-130} - q^{-132} + q^{-134} - q^{-136} - q^{-138} - q^{-142} - q^{-146} -2 q^{-148} - q^{-152} - q^{-156} - q^{-160} - q^{-162} - q^{-164} - q^{-166} +2 q^{-168} -2 q^{-170} + q^{-172} + q^{-178} - q^{-180} + q^{-182} - q^{-184} + q^{-186} - q^{-190} + q^{-192} + q^{-196} }[/math]

.

Computer Talk
The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`
Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.

Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:= K = Knot["9 3"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ 2 t^3-3 t^2+3 t-3+3 t^{-1} -3 t^{-2} +2 t^{-3} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ 2 z^6+9 z^4+9 z^2+1 }[/math]
In[6]:= Alexander[K, 2][t]
KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.
Out[6]= [math]\displaystyle{ \{1\} }[/math]
In[7]:= {KnotDet[K], KnotSignature[K]}
Out[7]= { 19, 6 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ -q^{12}+q^{11}-2 q^{10}+3 q^9-3 q^8+3 q^7-2 q^6+2 q^5-q^4+q^3 }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ z^6 a^{-6} +z^6 a^{-8} +5 z^4 a^{-6} +5 z^4 a^{-8} -z^4 a^{-10} +6 z^2 a^{-6} +7 z^2 a^{-8} -4 z^2 a^{-10} + a^{-6} +3 a^{-8} -3 a^{-10} }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ z^8 a^{-8} +z^8 a^{-10} +z^7 a^{-7} +2 z^7 a^{-9} +z^7 a^{-11} +z^6 a^{-6} -5 z^6 a^{-8} -5 z^6 a^{-10} +z^6 a^{-12} -4 z^5 a^{-7} -8 z^5 a^{-9} -3 z^5 a^{-11} +z^5 a^{-13} -5 z^4 a^{-6} +9 z^4 a^{-8} +11 z^4 a^{-10} -2 z^4 a^{-12} +z^4 a^{-14} +3 z^3 a^{-7} +9 z^3 a^{-9} +4 z^3 a^{-11} -z^3 a^{-13} +z^3 a^{-15} +6 z^2 a^{-6} -9 z^2 a^{-8} -11 z^2 a^{-10} +3 z^2 a^{-12} -z^2 a^{-14} -4 z a^{-9} -z a^{-11} +z a^{-13} -2 z a^{-15} - a^{-6} +3 a^{-8} +3 a^{-10} }[/math]

Vassiliev invariants

V2 and V3: (9, 26)
V2,1 through V6,9:
V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9
[math]\displaystyle{ 36 }[/math] [math]\displaystyle{ 208 }[/math] [math]\displaystyle{ 648 }[/math] [math]\displaystyle{ 1578 }[/math] [math]\displaystyle{ 246 }[/math] [math]\displaystyle{ 7488 }[/math] [math]\displaystyle{ \frac{39616}{3} }[/math] [math]\displaystyle{ \frac{7072}{3} }[/math] [math]\displaystyle{ 1744 }[/math] [math]\displaystyle{ 7776 }[/math] [math]\displaystyle{ 21632 }[/math] [math]\displaystyle{ 56808 }[/math] [math]\displaystyle{ 8856 }[/math] [math]\displaystyle{ \frac{1125053}{10} }[/math] [math]\displaystyle{ \frac{16214}{5} }[/math] [math]\displaystyle{ \frac{218942}{5} }[/math] [math]\displaystyle{ \frac{1313}{2} }[/math] [math]\displaystyle{ \frac{57373}{10} }[/math]

V2,1 through V6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V2 and V3.

Khovanov Homology

The coefficients of the monomials [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s-1 }[/math], where [math]\displaystyle{ s= }[/math]6 is the signature of 9 3. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 0123456789χ
25         1-1
23          0
21       21 -1
19      1   1
17     22   0
15    11    0
13   12     1
11  11      0
9  1       1
711        0
51         1
Integral Khovanov Homology

(db, data source)

  
[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=5 }[/math] [math]\displaystyle{ i=7 }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=9 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

[math]\displaystyle{ \textrm{Include}(\textrm{ColouredJonesM.mhtml}) }[/math] 

In[1]:=    
 
<< KnotTheory`
 
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=
Crossings[Knot[9, 3]]
Out[2]=  
9
In[3]:=
PD[Knot[9, 3]]
Out[3]=  
PD[X[8, 2, 9, 1], X[12, 4, 13, 3], X[18, 10, 1, 9], X[10, 18, 11, 17], 

 X[14, 6, 15, 5], X[16, 8, 17, 7], X[2, 12, 3, 11], X[4, 14, 5, 13], 



  X[6, 16, 7, 15]]
In[4]:=
GaussCode[Knot[9, 3]]
Out[4]=  
GaussCode[1, -7, 2, -8, 5, -9, 6, -1, 3, -4, 7, -2, 8, -5, 9, -6, 4, -3]
In[5]:=
BR[Knot[9, 3]]
Out[5]=  
BR[3, {1, 1, 1, 1, 1, 1, 1, 2, -1, 2}]
In[6]:=
alex = Alexander[Knot[9, 3]][t]
Out[6]=  
     2    3    3            2      3

-3 + -- - -- + - + 3 t - 3 t  + 2 t



     3    2   t


     t    t
In[7]:=
Conway[Knot[9, 3]][z]
Out[7]=  
       2      4      6
1 + 9 z  + 9 z  + 2 z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{Knot[9, 3]}
In[9]:=
{KnotDet[Knot[9, 3]], KnotSignature[Knot[9, 3]]}
Out[9]=  
{19, 6}
In[10]:=
J=Jones[Knot[9, 3]][q]
Out[10]=  
 3    4      5      6      7      8      9      10    11    12
q  - q  + 2 q  - 2 q  + 3 q  - 3 q  + 3 q  - 2 q   + q   - q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{Knot[9, 3]}
In[12]:=
A2Invariant[Knot[9, 3]][q]
Out[12]=  
 10    14    18    20    22      24    30    32    34    36
q   + q   + q   + q   + q   + 2 q   - q   - q   - q   - q
In[13]:=
Kauffman[Knot[9, 3]][a, z]
Out[13]=  
                                          2       2       2      2

3    3     -6   2 z    z     z    4 z   z     3 z    11 z    9 z



--- + -- - a   - --- + --- - --- - --- - --- + ---- - ----- - ---- + 



10    8          15    13    11    9     14    12      10      8



a     a          a     a     a     a     a     a       a       a



    2    3     3       3      3      3    4       4       4      4
 6 z    z     z     4 z    9 z    3 z    z     2 z    11 z    9 z
 ---- + --- - --- + ---- + ---- + ---- + --- - ---- + ----- + ---- - 
   6     15    13    11      9      7     14    12      10      8
  a     a     a     a       a      a     a     a       a       a

    4    5       5      5      5    6       6      6    6    7
 5 z    z     3 z    8 z    4 z    z     5 z    5 z    z    z
 ---- + --- - ---- - ---- - ---- + --- - ---- - ---- + -- + --- + 
   6     13    11      9      7     12    10      8     6    11
  a     a     a       a      a     a     a       a     a    a

    7    7    8     8
 2 z    z    z     z
 ---- + -- + --- + --
   9     7    10    8


   a     a    a     a
In[14]:=
{Vassiliev[2][Knot[9, 3]], Vassiliev[3][Knot[9, 3]]}
Out[14]=  
{0, 26}
In[15]:=
Kh[Knot[9, 3]][q, t]
Out[15]=  
 5    7    7      9  2    11  2    11  3    13  3      13  4    15  4

q  + q  + q  t + q  t  + q   t  + q   t  + q   t  + 2 q   t  + q   t  + 



  15  5      17  5      17  6    19  6      21  7    21  8    25  9


  q   t  + 2 q   t  + 2 q   t  + q   t  + 2 q   t  + q   t  + q   t
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